Correcting Wealth-Scaled Downside Risk in Portfolio Optimization

# Why is this utility function not picking up its penalty?


# Why is this utility function not picking up its penalty?












I was reading this seminal paper by Infanger. On page 40, Figure 11. was quite interesting. In particular I was interested in the top one, 19 Years and I wanted to reproduce this plot. To give some background: It's about utility maximization which should be solved by DP approach, i.e.

$$ \max_{x_t,0\le t\le T} E[u(W_T)]$$

where $u$ is a utility function and $W_T$ is the wealth at time $T$. We want therefore to maximize terminal wealth. For the picture he uses the following "quadratic downside risk" function

$$ u(W) = W - \frac{\lambda}{2}\max{(0,W_d-W)^2}$$

where $W_d$ is a target amound and $\lambda$ a scaling parameter. As wealth evolves via $W_{t+1} = W_t \cdot\langle x_t, R\rangle$ where $x_t$ are the allocation and $R$ the return, he writes down the Bellman equation of this problem:

$$V_{t}(W_t) = \max_{x_t}E[V_{t+1}(W_t\cdot \langle x_t, R\rangle)|W_t]$$ since I'm just interested in the final step, we have $V_T = u$ and the maximization problem I want to solve is $$V(W) = \max_{x}E[u(W\cdot \langle x, R\rangle)|W] $$ dropping the time $t$ index. I assumed (as Infanger did if I get him right), that $R$ are normally distributed. I use then Gauss-Hermite quadrature to approximate the expectation (see page 71 in this paper). $$V(W) = \frac{1}{\sqrt{\pi}}\max_{x}\sum_{i=1}^m w_i u(W(1+\hat{\mu}(x)+\sqrt{2}\hat{\sigma}(x)\cdot q_i)) $$ where $\hat{\mu} = \langle \mu, x\rangle$ and $\hat{\sigma} = \sqrt{\langle x, \Sigma x\rangle}$ and $w_i$ are the Gauss-Hermite weights and $q_i$ the corresponding nodes.

I've coded a very simple and not optimized version to see if I get the desired picture.

first a picture of the utility function:

```
utility <- function(w){
  K <- 100000
  temp <- K-w
  temp[temp<=0] <- 0
  return(w - 1000*temp^2)
}

x <- seq(90000,120000,1000)
y <- utility(x)
plot(x,y,type="l")
```

Now I just generated a sequences of $W$ and solved the above problem. The following code junk defines the covariance and expected return vector. The data is from the Infanger paper above.

```
wealth <- seq(50000,150000,5000)
mu <- c(0.108, 0.1037, 0.0949, 0.079, 0.0561)
cor <- matrix(c(1, 0.601, 0.247, 0.062, 0.094,
                0.601, 1.0, 0.125, 0.027, 0.006,
                0.247, 0.125, 1.0, 0.883, 0.194,
                0.062, 0.027, 0.883, 1.0, 0.27,
                0.094, 0.006, 0.194, 0.27, 1.0),
              ncol=5, nrow=5,byrow=T)
std <- c(0.1572, 0.1675, 0.0657, 0.0489, 0.007)
temp <- std%*%t(std)
cov <- temp*cor
```

With this data at hand and a sequence of wealth (see above) I just run an optimization for each given wealth and store the solution (assuming no short selling). To solve the problem I used the `Rsolnp` package in R. It solves a minimization problem that's why I'm returning a $-1$ in the objective function below:

```
library(Rsolnp)
library(statmod)
obj <- function(x, currentWealth, mu, cov, r=0, nodes, weights){
  drift <- sum((mu-r)*x)+r
  cor <- sqrt(sum(x*(cov%*%x)))
  term1 <- currentWealth*(1+drift)
  term2 <- currentWealth*sqrt(2)*cor*nodes
  return(-1/sqrt(pi)*sum(weights*utility(term1+term2)))
}

g_constraints <- function(x,currentWealth, mu, cov, r=0, nodes, weights){
  return(sum(x))
}

x0 <- rep(0.25,length(mu))
weights <- gauss.quad(10,"hermite")$weights
nodes <- gauss.quad(10,"hermite")$nodes
solmat <- matrix(NA, ncol=length(mu),nrow=length(wealth))
for(i in 1:length(wealth)){
  sol <- solnp(pars=x0, fun = obj,
               eqfun = g_constraints,
               eqB = 1,
               LB = rep(0, length(mu)),
               UB = rep(1, length(mu)),
               currentWealth = wealth[i], mu = mu, cov = cov,
               r = 0, nodes = nodes, weights = weights)
  solmat[i,] <- sol$pars
  x0 <- sol$pars
}
colnames(solmat) <- c("US Stock", "Int Stocks", "Corp Bonds", "Gvnt Bond", "Cash")
rownames(solmat) <- as.character(wealth)
```

However, I get a constant allocation where all money is invested in US Stocks. What's wrong with this and how do I get this chart from Infanger?

## Answer by math (score 1, accepted)

https://quant.stackexchange.com/a/35113

The problem was a missing $W_t$ in the equation for correlation. I've updated the above code and did a rerun. We have now the following allocation which is much closer to the Infanger paper.

```
> solmat
           US Stock  Int Stocks Corp Bonds    Gvnt Bond         Cash
50000  5.043872e-01 0.089871441 0.40574133 2.745030e-08 1.788550e-09
55000  4.050341e-01 0.090625580 0.50434024 2.744996e-08 1.788417e-09
60000  3.222272e-01 0.091347143 0.58642565 2.744972e-08 1.788325e-09
65000  2.521815e-01 0.091750138 0.65606829 2.744945e-08 1.788218e-09
70000  1.920722e-01 0.092167629 0.71576010 2.744928e-08 1.788152e-09
75000  1.401952e-01 0.092551771 0.76725296 2.744917e-08 1.788109e-09
80000  9.542976e-02 0.092965569 0.81160464 2.744770e-08 1.787578e-09
85000  5.926248e-02 0.085949047 0.77256462 2.548693e-10 8.222386e-02
90000  2.556086e-02 0.042435548 0.35690767 2.546768e-10 5.750959e-01
95000  5.666367e-07 0.007460414 0.02786258 1.724982e-12 9.646764e-01
1e+05  4.086260e-03 0.018524853 0.14238886 1.318764e-04 8.348682e-01
105000 4.229705e-03 0.021298108 0.33601246 1.319004e-04 6.383278e-01
110000 4.261748e-03 0.022020057 0.49978208 1.319047e-04 4.738042e-01
115000 1.014474e-02 0.042426599 0.62859439 3.219158e-03 3.156151e-01
120000 1.040348e-02 0.046451692 0.76095787 3.218435e-03 1.789685e-01
125000 1.308464e-02 0.132081249 0.79793418 3.218537e-03 5.368139e-02
130000 1.429210e-02 0.239570571 0.72417454 3.105067e-03 1.885772e-02
135000 1.471366e-02 0.313967841 0.65235180 3.064252e-03 1.590244e-02
140000 1.492658e-02 0.369961444 0.59727148 3.044485e-03 1.479601e-02
145000 1.506062e-02 0.416426483 0.55128612 3.032353e-03 1.419442e-02
150000 1.515564e-02 0.456708072 0.51130955 3.023891e-03 1.380285e-02
```